Question

Solve the equation \(2 \mathrm{x} /(3+\mathrm{x})+(3+\mathrm{x}) / 3=2+\mathrm{x}^{2} /[3(\mathrm{x}-3)\)

Short answer

The solution to the equation \(2x/(3+x) + (3+x)/3 = 2 + x^2/[3(x-3)]\) is \(x = 0\).

Step-by-step solution

Find the Least Common Denominator (LCD)

The equation contains the following denominators: (3+x), 3, and 3(x-3). The LCD will be the product of these distinct factors. Therefore, the LCD is: LCD = 3(3 + x)(x - 3)

Distribute the LCD to Each Term and Simplify

Multiply each term by the LCD to eliminate the fractions: \(3(3 + x)(x - 3)[ \frac{2x}{3 + x} + \frac{3 + x}{3} ] = 3(3 + x)(x - 3)[2 + \frac{x^2}{3(x - 3)}]\) This simplifies to: \(2x(3)(x - 3) + (3 + x)(3 + x)(x - 3) = (3 + x)(2)(x - 3) + x^2(3 + x)\)

Rearrange the Equation and Solve for x

Now, we expand and simplify the equation: \(6x^2 - 18x + 3x^2 - x^3 = 6x^2 - 18x + 3x^2 + x^3\) Combine like terms: \(x^3 + 3x^2 - x^3 = 6x^2 - 18x\) This simplifies further to: \(3x^2 = 6x^2 - 18x\) Now, we can subtract 3x^2 from both sides to get: \(3x^2 - 3x^2 = 6x^2 - 3x^2 -18x\) This simplifies to: \(0 = 3x^2 - 18x\) Factor out a 3x from the right side: \(0 = 3x(x - 6)\) Now, we can set each factor equal to zero and solve for x: 3x = 0, x = 0 x - 6 = 0, x = 6

Check the Solutions

We now have two possible solutions, x = 0 and x = 6. We must check each solution in the original equation to verify if they are valid. For x = 0: \(2(0)/(3+0) + (3+0)/3 = 2 + 0^2/[3(0-3)]\) \(0/3 + 3/3 = 2 + 0/-9\) \(0 + 1 = 2 + 0\) \(1 = 1\) (True) For x = 6: \(2(6)/(3+6) + (3+6)/3 = 2 + 6^2/[3(6-3)]\) \(12/9 + 9/3 = 2 + 36/9\) \(4/3 + 3 = 2 + 4\) \(7/3 ≠ 6\) Thus, based on our check, only the solution x = 0 is valid.

Key concepts

Least Common Denominator (LCD)

When solving rational equations, finding the Least Common Denominator (LCD) is crucial. This process helps to eliminate fractions, simplifying the equation to a form that is easier to work with. To find the LCD, look at all the denominators in the equation. Each denominator might contain multiple factors, so list them all. In this case, the denominators were \(3+x\), \(3\), and \(3(x-3)\). The LCD is formed by finding the product of all these distinct factors.

• List each unique factor from every denominator.
• For our equation, factors are \(3\), \(3+x\), and \(x-3\).
• The LCD is \(3(3+x)(x-3)\).

Using the LCD allows you to multiply every term in the equation. This will eliminate the fractions by ensuring each term of the equation has the same denominator, making the equation easier to solve.

Factoring

Factoring is a method used to rewrite an expression as a product of its factors. In the context of solving equations, factoring is often the step needed to break down complex polynomials into simpler components. For this exercise, after finding and multiplying by the LCD, the next step involved simplifying the expression. This allowed us to rearrange the equation into terms that can be combined.

• Identify groups of terms that can be factored.
• In our example, the expression \(3x^2 - 18x\) was factored as \(3x(x - 6)\).
• Set each factor equal to zero to solve for potential solutions.

Factoring is an essential skill in algebra, as it breaks down expressions into their simplest forms, allowing for easier manipulation and solution finding.

Checking Solutions

Once you've found potential solutions, it's important to verify their validity. This step ensures that no extraneous solutions—solutions that emerge through the manipulation process but do not satisfy the original equation—are included in the final answer. To check, simply substitute the solutions back into the original equation.

• For each solution, replace \(x\) in the original equation with the solution.
• Calculate both sides of the equation to ensure they are equal.

In our particular problem, we found \(x = 0\) and \(x = 6\) as potential solutions. However, only \(x = 0\) made both sides of the equation equal, confirming it as the true solution. Always respect this step to avoid including incorrect solutions.

Equation Simplification

Simplifying an equation means reducing it to its simplest form so it can be more easily solved. This process often involves distributing terms, combining like terms, and eliminating unnecessary complexities. In this scenario, after the fractions were removed by using the LCD, it was crucial to expand and simplify both sides of the equation.

• Distribute any multipliers through the terms in parentheses.
• Combine like terms to collect variables and constants together.

For example, after multiplying by the LCD, terms were rearranged and combined to form \(3x^2 - 18x\). Ensure to follow each simplification step methodically, to not lose vision of how it affects finding the solution.